Whilst it might appear complex and lack the intuitiveness of the
"Colour Balance and Levels" tools "Curves" is a lot
more powerful and infinitely more flexible than any of the adjustment
tools discussed previously.

Figure 1
Figure 1 shows the default display for "Curves" with the 4 by 4
grid representing the 1/4, 1/2 and 3/4 tones. For many Photoshop users
this configuration is perfectly adequate. For those who prefer to work
with a finer grid, which broadly mirrors the Zone System used in B&W
darkroom printing circles we can hold down the ALT/OPTION key click the
mouse anywhere on the 4 by 4 grid and a 10 by 10 grid will appear.
Repeating this procedure will cause the grid to toggle between the 4 by 4
grid and 10 by 10.
"Curves" operate in much the same way as "Levels" but
have added benefits. The following screenshot will produce the same
results as dragging the "Levels Input" sliders towards the centre
causing and overall increase in image contrast.

Figure 2
The horizontal axis represents Input values and the vertical
axis Output values. Setting the Input value at level 50 to
Output value 0 has the effect of clipping every pixel between
level 0 and 50 to level 0; thus ALL pixels in that range will become
totally black. Setting Input level 200 to Output level 255
means that all pixels in the range 200 to 255 will become white. The
following image is not a pretty sight but it should give you some idea as
to what is happening in the above example. Notice that sections of cloud
are now completely without detail. The trees towards the left edge of the
image have now become black.

Image after Curve in Figure 2
has been applied
Dragging the diagonal line up and left has the same effect on the
image as dragging the Levels Gamma slider to the left (brightens "midtones").
Whilst dragging the diagonal down and right will cause the "midtones" to
darken.
Figure 3 below is effectively the same as dragging the "Levels
Output" sliders towards the centre and will result in the overall
contrast of the image being reduced.

Figure 3

Image after Curve in Figure 3
has been applied
Adjustments such as those that follow cannot be replicated by any
other Photoshop adjustment tool, which is why "Curves" tend to be
the adjustment tool of choice for more advanced users.
Figure 4 demonstrates an adjustment known as the "S-curve".
This particular adjustment increases image contrast in the "midtones"
by compressing the "shadows" and "highlights".
Notice the two black dots on the curve; there can be up to 14 of them
on the grid. These are sometimes called lock-down points and can be used
to define the shape and steepness of the curve. We can think of a steep
curve as high contrast and a shallow curve as low contrast.

Figure 4

Image after Curve in Figure 4
has been applied
Note that no colour correction has been undertaken for the above three
images.
The final screenshot in this series is another variation of the "S-curve".
This time we are reducing "midtone" contrast whilst expanding the
"highlights" and "shadows".

Figure 5
The above screenshots all demonstrated various possibilities for "tone
control" but "Curves" is the master when it comes to colour
correction. Just as we can select individual colour channels in "Levels"
we can do the same in "Curves". There really are no limits to the
number of possibilities. We can stretch and pull the diagonal line in any
direction, fix points pretty much where we want. However, be warned that
"Curves" are like an elastic band, and just as an elastic will
eventually break "Curves" will do likewise resulting in some weird
and wonderful/awful effects.
For a more comprehensive discussion on the "Curves" tool see
Real World Photoshop 6/7/CS by Bruce Fraser
and David Blatner or Photoshop 6/7/CS Artistry by Barry Haynes and
Wendy Crumpler.